# Projection of matrix onto subspace

I am confused about why the orthogonal projection of matrices onto subspaces is given by a change-of-basis-like formula. For example, in the below image from these notes, why is the orthogonal projection of matrix A onto the subspace $$V_m$$ given by $$V_m^* A V_m$$?

When I search online, I come across a formula like (for example, from the Wikipedia article), which does not look at all like $$V_m^* A V_m$$.

• What's written in the Wikipedia article is the projection matrix. $P_A$ is the matrix that projects a given vector onto the subspace spanned by the columns of $A$. Commented Feb 18, 2021 at 13:00

## 1 Answer

I have the same question, but don't have the reputation to comment.

It's worth noting that you have two different $$A$$ matrices in your question - the $$A$$ in the standard projection formula corresponds to your $$V_m$$. Because the column-vectors of the subspace are orthonormal, $$V_m^T V_m = I$$, and so the projection matrix (in this notation) is $$P \equiv V_m V_m^T$$. Here is where I get stuck.